no  ?. 


California.   University.    Department 
of  Tnechan5cal   engirt 
•:  iletln. 


no 


UNIVERSITY  OF  CALIFORNIA 
AT   LOS  ANGELES 


NO. 


qjartment  of  Jjedtmtkal  Jnginwing, 


UNIVERSITY  OK  CALIFORNIA. 


HYDRAl'LIC  STEP. 


BERKELEY: 


BULLETIN   No 


Iqjartment  of  jkhanical  tnjjtnemng, 


UNIVERSITY   OF   CALIFORNIA. 

» 


HYDRAULIC  STEP. 


BERKELEY: 

1887. 


Analysis  of  a  Hydraulic  jStep 

The  machinery  which  was  used  in  connection  with  a  cer- 
tain pumping  system  for  irrigation  contained  a  vertical  shaft 
which  supported  a  heavy  load  and  rotated  in  water  at  a  high 
velocity.  The  whole  plant  was  required  to  be  of  such  con- 
struction as  to  dispense  with  the  necessity  of  supervision. 

Under  these  conditions  the  footstep  of  the  shaft  claimed 
careful  consideration.  Lubrication,  except  with  water,  was 
out  of  the  question.  The  use  of  lignum  vitse  was  also  of 
doubtful  utility  on  account  of  the  grit,  of  which  the  water 
carried  considerable  quantities.  Girard's  method  of  forcing 
water  between  the  surfaces  of  contact  by  means  of  a  force- 
pump*  was  too  complicated  and  lacked  permanency. 

In  this  dilemma  it  occurred  to  me  that  a  permanent  water 
pressure  might  be  established  through  the  agency  of  a  forced 
vortex,  created  by  the  rotation  of  the  shaft  itself,  which,  in 
acting  against  a  disc  attached  to  the  shaft,  could  be  made  to 
balance  a  portion  of  or  the  whole  load. 

Fig.  1  is  a  sectional  view  of  the  footstep,  through  the  axis 
of  shaft,  S.  It  represents  a  cylindrical  vessel,  composed  of  a 
number  of  identical  compartments  m,  m,  etc.  (the  drawing 
shows  three).  The  concentric  shaft  S,  passes  freely  through 
these  and  rests  on  an  ordinary  footstep,  which  is  attached  to 
the  bottom  of  the  lowest  compartment.  The  shaft  carries  for 
each  division  a  disc  of  a  diameter  just  sufficient  to  clear  the 
surrounding  face  of  the  cylinder,  and  midway  between  top 
and  bottom.  The  upper  part  of  each  disc  is  provided  with 
radial  ribs  or  blades  r,  T,  and  the  bottom  of  each  vessel  is  pro- 
vided with  stationary  ribs  r,  r. 

*See  Armengaud,  "  Vignole  des  Mecaniciens,"  p.  13,  and  Note  "Sure  lea 
Experiences  de,"  etc.,  in  Comptes  Rendues  de  1'Academie  dea  Sciences  a 
Paris,  T.  55. 

41S65O 


For  our  purpose,  and,  as  will  be  seen  also  in  common  prac- 
tice, it  is  sufficient  to  consider  only  one  compartment.  If  the 
shaft  rotates  and  the  vessel  is  filled  with  water,  the  wings,  r, 
will  cause  the  latter  to  rotate  with  the  same  angular  velocity. 
A  forced  vortex  is  created,  and  the  pressure  of  the  water  will 
increase  from  the  center  towards  the  circumference.  The 
water  in  the  space  below  the  rotating  disc  will  be  under  a 

Fig.  1 


,' 


uniform  pressure,  equal  to  the  maximum  pressure  above; 
hence  the  disc,  and  with  it  the  shaft,  will  be  under  the  influ- 
ence of  an  upward  pressure,  equal  to  the  difference  of  the  total 
pressures  acting  upon  both  sides  of  the  disc. 

Let  67  denote  the  angular  velocity  of  rotation  ; 

r,  the  radius  of  the  rotating  disc  in  feet; 

r0,  the  radius  of  the  hub  in  feet; 

n,  the  number  of  revolutions  per  minute; 

v,  velocity  of  rotation  at  any  given  distance; 


v0,  the  velocity  of  rotation  at  r,; 

y,  the  acceleration  of  gravity; 

y,  the  density  of  the  fluid; 

m,  the  number  of  rotating  discs; 

PI,  the  total  pressure  upon  the  upper  face  of  the  disc; 

P2,  the  total  pressure  upon  the  lower  face  of  the  disc; 

P=P.i — PI,  the  resulting  upward  pressure. 
Assuming  r0  to  be   zero,  and   neglecting  fluid   friction,  we 
may  obtain  the  pressure  P  as  follows: 

The  hight  to  which  the  fluid  would  rise  at  a  distance    x> 
from   the   axis,    measured   from    the   plane   bVb,  is  equal  to 

v*      &?V 
2~~-$ — •    The  ordinates  representing  these  heads  terminate, 

therefore,  in  the  surface  of  a  paraboloid  of  revolution,  with  its 
vertex  at  V,  and  its  volume   aVa,  is  equal  to  one-half  that  of 

the  cylinder  baab,  hence  Pi=y  n  r*T~>  and  considering  that 

the   maximum    specific   pressure,    y  r 2  r—  acts  upon  the  en- 

u 

tire  lower  face  of  the  disc,  we  have  P2=y  n  r*   — ,  hence 


r *g 

For  any  other  value  of  r0  we  find 

Pi=y  n  (?t2 — ?'o)8  T~  (see  notie  below), 

and  since  the  maximum  specific  pressure  acts  upon  the  entire 
area  of  the  disc,  we  have: 

P,«r*(r*^J)r»~,aad 

P=y7f  (r4— r^)~- 

If  we  express  oo  in  terms  of  n  and  r,  r0,  in  terms  of  d,  da 
(diameters),  we  find: 

P=0.00104  (d*—d*)  n*,  and  for  m  rotating  discs 
P=0.00104  (d*—d*}  n*  m. 


NOTE. — The  specific  pressure  p,  at  distance  x,  is  equal  t«  y  (a? — rj)  

a?2  rr  ca2 

jnce  Pi      1  ity  —  |       (x2— r=)  x  dx=n y  (r2— r=) 2  y^ ' 


The  work  in  the  laboratory  has  to  deal  with  two  subjects  of 
investigation,  viz. :  Pressure,  and  work  consumed  by  f ric- 
tional  resistance  of  the  disc  rotating  in  water. 

The  value  for  pressure,  found  by  the  above  formula, 
gives  the  same  result  as  that  found  in  practice,  provided  fluid 
friction  does  not  exist;  its  presence"  will  not  only  decrease  the 

maximum  pressure,  y  T*  — ,  but  also  the  total  pressure  under 

u 

the  disc,  which  is  now  no  longer  the  resultant  of  a 
uniform  maximum  pressure  under  the  disc.  The  cause  of  this 
is  to  be  found  in  the  clearance  or  space  between  the  movable 
and  stationary  ribs  and  surfaces. 

The  rotating  ribs  do  not  impart  the  same  motion  to  all  the 
water  above  the  disc.  There  remains  a  film  of  fluid  next  to 
the  upper  stationary  surface,  either  wholly  stationary,  or  if 
slippage  takes  place,  partially  so.  Within  the  free  space,  or 
clearance,  inner  friction  causes  the  water  to  assume  gradually, 
according  to  some  law,  the  full  velocity  of  rotation.  The 
pressures  due  to  the  normal  forces  being  thus  modified,  radial 
currents  towards  the  axis  of  rotation  are  induced,  while  a 
compensating  flow  from  the  axis,  in  conformity  to  the  law  of 
continuity,  takes  place  between  the  rotating  ribs.  In  other 
words,  potential  energy  is  converted  into  kinetic  energy. 
The  same  action,  but  with  currents  reversed,  takes  place  in 
the  compartment  below  the  disc. 


The  problem  mapped  out  in  the  foregoing  was  given  to  the 
students  as  laboratory  work.  It  calls  for  mechanical  appli- 
ances, instruments  of  precision  for  measurement,  etc.,  of 
varied  character;  it  appeals  to  their  knowledge  acquired  in 
the  lecture-room  to  reach  correct  conclusions,  and  the  results 
are  not  merely  of  temporary  value  but  find  important  appli- 
cations. 

The  students  made  the  experiments  and  assisted  in  the  de- 
signing and  drawing  of  the  apparatus,  all  of  which  was  built, 
with  the  assistance  of  the  students,  by  Jos.  A.  Sladky,  Super- 
intendent of  the  Machine  Shops.  His  skill  and  ready  com- 
prehension of  the  cardinal  points  aimed  at,  have  largely  con- 
tributed to  the  successful  results  of  the  experiments. 


Experiments  for  Pressure. 

The  footstep  constructed  to  suit  experimental  convenience  is 
shown  in  Fig .  2b ,  which  represents  a  section  through  the  axis 
of  the  shaft.  The  inner  diameter  of  the  cylinder  is  12£ 
inches  in  order  to  admit  a  12-inch  disc;  but  discs  of  less  diam- 
eter can  be  inserted  together  witli  corresponding  rings  V, 
and  stationary  ribs  R',  R. 

The  shaft  >S",  passes  through  another  shaft,  S,  which  has  its 
bearings  in  the  hub  F,  of  the  frame,  and  on  which  are 
mounted  the  driving  pulleys  P,  P.  Rotation  is  communi- 
cated to  S'  by  means  of  a  couple  G.  In  this  way  the  tension 
of  the  belt  is  entirely  taken  up  by  the  shaft  S,  and  its  bear- 
ings F,  and  since  the  resisting  forces  upon  the  disc  occur 
only  in  couples,  the  shaft  S',  is  left  free  to  slide  in  its  bearings 
without  friction. 

The  upper  end  of  the  shaft  S',  engages  one  end  of  the 
balance  lever  L,  through  the  medium  of  the  socket  or  step 
D,  which  is  prevented  from  rotating,  as  seen  in  the  figure, 
in  order  to  protect  the  pointed  pivot  which  acts  upon  L.  The 
other  end  of  lever  L,  is  supported  by  the  platform  of  a  scale 
C,  Fig.  2,  which  measures  the  pressure  upon  the  disc. 

The  experimental  data  are  given  in  the  following  table,  in 
which 

d  denotes  the  diameter  of  the  disc  in  inches, 

n       "          '    number  of  revolutions  per  minute, 

R      "         "    reading  of  the  scale, 

</  ' '     weight  of  block  on  scale  which  engages  lever  L. 

gt      "          "    weights  of  shaft  S',  and  socket  7), 

gt      "         "    weight  of  disc  in  water, 

m     "          "     reaction  of  level*  L  on  shaft, 

mt    "  "        "       "       "       platform, 

P     "         "    pressure  of  water  upon  the  disc. 

The  ratio  of  the  lever-arms  is  2  :  1. 

We  have  P=212+^-f-0/  +  ?n/— 20— w. 


10 


P 

Ratio  of 

P                 i  Ratio  of 

MM        -^      _             !    nVninnrnrl 

d 

n 

Computed 

Observed. 

to    Com- 
puted P. 

d 

n 

Computed 

Observed. 

to    Com- 
puted P. 

5" 

1510     71.5 

64.9 

090 

9"  1786    2056 

1861 

091 

1128     382 

365 

0.96 

626     130.2 

1301 

1.00 

710     15.8 

147 

093 

|440|      64.3 

61.7 

096 

557:      9.8 

9.1 

093 

2«->4|      21.2 

20.1 

0.95 

7" 

1155  160.5 

148.6 

0.93 

12"  462|    222  8 

2208 

0.98 

932  105.7 

96.1 

0.92 

443     204.1 

197.8 

0.97 

764    70.2 

668 

0.95 

300;      94.7      i      95.1           1.00 

424     21  6 

22.2 

1.03 

252      680            65.6      i      0.96 

284       9.7            10.2 

1.05 

186;      21  3 

20.1      j      0.94 

1                 i 

1 

Combining  these  results,  we  find  the  ratio  for  the  several 
discs: 

For  5"   disc  0  926 

adopted  value,  0.95. 


I      H     °  U*J  ^ 
12      "      0973J 


Multiplying  .95  into  the  coefficient  of  P   (see  page  5),  we 
find:  P=0.001  (d*—  d*}  n*  m. 


Resistance. 

The  arrangement  as  represented  in  Fig.  2,  which  was  de- 
signed to  measure  pressure  and  resistance  at  the  same  time, 
would  have  necessitated  the  elimination  of  the  friction  mo- 
ment of  the  upper  footstep  of  shaft  S',  which  is  great,  con- 
sidering the  pressure  to  which  the  step  is  exposed.  For  this 
reason  the  experiments  for  determining  the  resistance  were 
made  independently  of  those  for  pressure,  which  made  it  pos- 
sible to  eliminate  the  latter  by  arranging  the  ribs  in  the  same 
way  both  under  and  above  the  disc.  The  resulting  resistance 
of  the  footstep  would  therefore  be  equal  to  one-half  of  the  sum 
of  the  two  resistances  found  for  movable  and  stationary  ribs 
on  each  side  of  the  disc. 

Fig.  3  represents  a  vertical  section  of  the  dynamometer, 
which  was  designed  and  built  for  these  tests. 

Fig.  4  is  a  front  view  and  Fig.  4a  a  top  view  of  the  index 
plate. 


11 

The  two  journal  bearings  J  J,  support  the  shaft  S',  which 
carries  the  pulley  P,  the  graduated  disc  D,  and  the  cylin- 
drical casing  C.  This  shaft  S',  is  bored  to  receive  the  shaft 
S,  shown  in  dotted  lines.  Connection  between  the  two  is 
established  through  the  medium  of  a  spiral  spring  t,  the  ends 
of  which  are  attached  to  the  box  C,  and  shaft  S,  respectively. 


Fig.  3 


The  lower  end  of  S  is  provided  with  a  disc  u,  which  car- 
ries two  coupling  pins  for  the  purpose  of  conveying  rotation 
to  the  driving  pulley  for  the  footstep  A,  Fig.  2.  The  relative 
position  of  the  two  shafts,  that  is,  the  tension  of  the  spring  t, 
measures  the  moment  required  to  communicate  motion  to  the 
footstep.  Since  precision  of  measurement  was  aimed  at  in  the 
construction  of  the  dynamometer,  the  recording  apparatus  was 


12 

not  designed  for  reading  while  in  motion,  because  such  an  ar- 
rangement would  have  impaired  its  sensitiveness. 

The  bar  x,  which  carries  the  index  hand  (see  Fig.  4a),  rests 
on  the  plate  D,  and  is  pivoted  on  the  shaft  S'.  n,  n,  are  two 
springs  attached  to  x  ;  their  outer  ends  are  bent  over  as  seen 
in  the  drawings.  If  these  springs  are  forced  down  so  as  to 
bear  upon  x,  their  outer  ends  enter  the  forked  ends  of  the 


arms  A,  A,  which  are  connected  with  the  disc  u  (see  Fig.  4). 
If  released,  they  become  disengaged  from  A,  A,  but  bind  x  to 
the  disc  D  (see  Fig.  3).  L,  L  are  two  levers,  pivoted  to  the 
bar  x.  In  the  position  in  Fig.  4,  their  short  ends  C,  C,  abut 
against  the  edge  of  a  slot  cut  in  the  springs  n,  n,  and  keep 
them  down.  By  a  slight  depression  of  L,  L,  however,  the 
arms  C,  C,  will  release  their  hold,  pass  through  the  slots  and 
the  springs  will  fly  up  (see  Fig.  3). 


13 

At  any  time,  during  the  action  of  the  dynamometer,  the 
position  of  the  index  hand  may  be  secured  by  pressing  upon  a 
knob  y  of  a  rod,  which,  by  means  of  lever  z,  sleeve  S 
and  levers  L,  disengages  C,  C.  During  the  use  of  the  dynam- 
ometer the  arms  A,  of  shaft  S,  are  connected  with  the  spring, 


plate  x,  and  index  hand,  Fig.  4,    which   relative   position   is 
secured  as  explained. 

Resistance  Determined  by  Dynamometer. 

Let  a=  index  reading  of  dynamometer. 

o-1=index  reading  of  dynamometer  without  water   in 
the  footstep,  to  eliminate  frictional  resistance  and  the 
correction  for  zero  point  of  the  graduated  disc. 
M =corrected  moment  of  resistance  of  footstep. 
f_0. 02298  value  of  one  division  of  index  plate. 
r= radius  of  pulley  on  footstep  shaft. 
r!=radius  of  pulley  on  dynamometer. 

r  =0.494 
r, 

Then  we  have  M=er   (a— «I)=0.01135  («—«,). 


14 

One  hundred  and  thirty-four  tests  for  moment  of  resistance 
of  movable  and  stationary  wings,  were  made. 

The  motive  power  was  derived  from  a  small  Pelton  wheel, 
but  the  water  supply  proved  insufficient  to  obtain  the  desired 
velocities.  The  very  laborious  computations  by  the  method 
of  least  squares  also  established  the  fact  that  the  results  could 


not  be  trusted  on  account  of  the  unavoidable  slipping  of  the 
belts. 

For  these  reasons  I  concluded  to  apply  a  different  method, 
applicable  not  only  for  the  step,  which  was  the  original  object, 
but  in  general  for  the  determination  of  fluid  resistance  to  ro- 
tating discs,  cylinders,  etc.  . 


IS 


Resistance  Measured  by  Read  ion. 

The  apparatus  built  for  this  purpose  is  shown  in  Fig.  5  in 
vertical  section. 

M,  M,  represents  a  closed  cylinder,  15"  by  7",  in  two  sec- 
tions bolted  together,  accurately  turned  and  balanced  and 
resting  on  a  conical  pivot  U  which  is  adjustable  by  a 
set  screw.  The  shaft  8  rests  upon  an  inverted  conical  pivot 
attached  to  cylinder  M,  and  has  its  upper  bearing  in  a  sleeve, 


Pig.  6 


which  is  inserted  in  the  hub  of  the  frame  M.  This  shaft  is 
provided  with  a  shoulder  and  set  screws  for  attaching  objects 
to  be  experimented  with.  D,  D,  represent  plates  or  discs 
capable  of  being  adjusted  to  any  position  required  for  the  test. 
The  upper  journal  bearing  of  the  shaft  passes  through  a 
central  opening  in  the  cylinder,  just  large]enough  not  to  touch. 
The  position  of  the  cylinder  is  maintained  by  four  friction 
rollers  R,  R. 


16 

The  frame  F  is  provided  with  bearings  for  a  shaft  S',  which 
carries  the  driving  pulley  P.  Motion  is  transmitted  to  shaft 
S  by  means  of  a  coupling  which  is  represented  in  Fig.  6  on  a 
larger  scale. 

The  disc  or  crosspiece  a  of  the  driving  shaft  S'  carries  two 
pins  p,  p.  The  disc  d  of  the  driven  shaft  S  has  pivoted  to  it 
at  c,  c,  two  levers,  which  engage  each  other  at  the  center  of 
the  shaft  r,  while  the  outer  ends  receive  the  thrust  of  the  pins 
p,  p,  at  i,  i. 

The  points  i  c  r  c  i,  and  also  the  centers  of  the  pins  p,  p,  are 
contained  in  diameters. 

It  is  evident  that  both~pins  bear  against  the  levers  with 
equal  pressure*  and  that  in  consequence  the  resultant  pressure 
transmitted  to  the  shaft  journals  is  zero. 

The  moment  of  resistance  is  measured  by  the  reaction  upon 
the  vessel  M.  The  rotation  of  this  vessel  is  not  influenced  by 
journal  friction  in  the  stationary  bearings,  but  the  point  to  be 
aimed  at  was  precision  of  motion — that  is,  the  avoidance  of 
vibrations  caused  by  bearings  enlarged  by  wear. 


Speed  Indicator. 

The  recording  apparatus  for  the  velocity  of  rotation  is  ap- 
plied at  the  upper  end  of  shaft  S'.  The  counter  in  common 
use  which  is  applied  at  the  end  of  the  shaft,  etc.,  could  not  be 
trusted,  as  the  possibility  of  slipping  and  the  source  of  error 
in  time-interval,  caused  by  a  lack  of  prompt  application  and 
withdrawal  of  the  instrument,  excludes  observations  for  small 
time-intervals,  which  become  tedious,  and  are,  under  certain 
conditions,  even  inadmissible. 

The  following  is  a  description  of  the  recording  apparatus 
designed  and  built  to  obviate  these  defects: 

A ,  N,  Figs.  8  and  9,  represent  two  identical  speed  indicat- 
ors, which  are  pivoted  at  0,  0.  In  the  position  shown  they 

*In  an  ordinary  pin  coupling  skill  may  produce  practically  an  equal  contact, 
but  an  almost  imperceptible  change  of  the  centers  of  the  shafts  will  produce 
an  unequal  distribution  of  the  pressures.  In  practice  these  centers  cannot 
remain  in  line,  hence  the  above  result.  In  the  coupling  described  the 
equality  of  pressure  is  not  interfered  with  by  such  derangement. 


17 

are  in  gear  with  a  pinion  P,  Fig.  8,  which  is  attached  to  the 
end  of  shaft  S'.  These  positions  are  maintained  by  springs 
8,  S,  and  checks.  The  position  which  corresponds  to  contacts 
of  the  armatures  A  with  their  electromagnets  M,  M,  leaves 
the  spur  wheels  and  pinion  P  out  of  gear.  The  contact  is 
maintained  by  a  projection  on  each  indicator,  which 

Fig.  7 


locks  behind  a  tooth  on  the  spring  bar  H  (see  Fig.  7). 
The  action  of  the  indicator  is  as  follows:  In  Fig.  9  both 
indicators  are  in  gear  with  pinion  P.  While  the  pointer  of 
the  pendulum  is  between  the  two  contact  springs,  and  just 
preceding  that  reading  of  the  dial-plate  of  the  clock,  from 
which  it  is  intended  to  count,  the  push  button  Rl  must  be 
depressed. 


18 

As  soon  as  the  pendulum  has  completed  its  stroke  and  pro- 
duced contact,  a  current  from  battery  B  passes  in  the  direc- 
tion of  the  arrows,  the  electromagnet  attracts  the  armature, 
and  the  indicator  N\  is  thrown  out  of  gear  against  the  spring 
and  retained  as  described.  Just  preceding  the  specified  in- 
terval of  time  as  before,  the  push  button  R2  must  be  depressed, 
which  causes  the  indicator  N  2  to  be  thrown  out  of  gear  and 
held  in  that  position  as  explained. 


Fig.  9 


The  difference  of  the  two  readings  gives  the  number  of  revo- 
lutions made  during  the  interval. 

The  result  is  liable  to  be  influenced  by  an  error  in  the  time 
intervening  between  the  closing  of  the  current  and  the  throw- 
ing out  of  gear  of  the  index.  If  the  time-interval  were  the 
same  for  both  indicators,  this  error  would  be  eliminated;  but 
such  can  hardly  be  expected  on  account  of  difference  in  fric- 
tion, distribution  of  masses,  intensity  of  electromagnets,  etc. 

This  error  will  influence  the  result  the  more,  the  smaller  the 
clock  interval  is.  We  can,  However,  from  observed  data, 


19 

compute  a  correction  which  will  make  it  possible  to  obtain  ex- 
cellent results  for  very  small  intervals. 

Let  the  time  intervening  between  the  closing  of  the  currents 
and  the  release  of  the  spur  wheel  from  the  pinion  be   repre- 
sented respectively  by  ^  and  £,,  for  counters  1  and  2. 
Let  i\  denote  the  first  reading  of  clock  ; 
r2  denote  the  second  reading  of  clock; 
T  denote  the  difference  of  these  readings  ; 
then  n  +  ^^rtimeat  instant  of  release  of  counter  1; 
and  r.2  +  £2=time  at  instant  of  release  of  counter  2. 
Hence,  r.2  +  t.2  —  (r1+*1)=2T+^  —  ^=true  time  interval,  which 
corresponds  to  the  difference  of  readings  of  the  dial. 

If  we  repeat  the  observations  for  the  same  T,  but  in 
order  reversed  regarding  the  push  buttons,  and  if  we  denote 
by  d  and  diy  respectively,  the  difference  of  dial  readings,  we 
have: 

T+  (tz  —  £,)  corresponding  to  d  ; 
T  —  (t.t  —  <j)  corresponding  to  d^ 

then  the  number  of  revolutions  n  per  second  equals 
d  d, 


Hence  t-t^T^-e  T 


Now,  if  we  always  make  one  observation  for  speed  in  the 
order  1  —  2  of  tiie  push  buttons,  we  find  the  true  number  of 

revolutions  per  minute  n=QO  ~ 

In  our  experiments  the  time-interval  T  was  generally  equal 
to  five  seconds  and  yielded  excellent  results. 

The  moments  were  measured  (see  Fig.  10)  by  weights  V,  V, 
attached  to  cords  which  fitted  a  groove  turned  on  the  flange 
of  the  cylinder  M.  Two  weights  were  used  to  counteract  the 
pressure  on  the  journal  U. 

In  order  to  relieve  the  friction  upon  the  step  pivot  U,  the 
cylinder,  etc.  ,  were  balanced  by  means  of  a  weight  W  and 
lever.  The  distance  between  the  points  of  suspension  and  the 
apparatus  was  over  14  feet,  so  that  practically  torsion  could 
not  be  felt.  The  upper  face  of  the  cylinder  was  adjusted 
horizontally  to  relieve  the  friction  rollers  R,  R,  from  lateral 
pressure. 


For  each  of  the  5,  7.  9  and  12  inch  discs,  two  sets  of  experi- 
ments were  made  to  ascertain  the  moment  of  resistance  for 
stationary  and  movable  wings.  The  resistance  offered  to  a 
solid  moving  in  a  fluid  is  practically  proportional  to  the 
square  of  the  velocity.  I  found  this  to  be  the  case  for  each 
individual  set  of  tests,  hence  we  have  for  the  same  disc 
M=fins,  M  representing  moment  of  resistance  and  ft  a  con- 


"^SMSSMS^^ 


w/////////////////wm^^ 


stant.     These  values  for  ft  were  computed  by  the  method  of 
least  squares,  and  are  given  in  the  following  table: 


d 

Number 

ft                   j  Number 

ft 

of 

of 

Inches. 

Observ. 

For  Stationary  Ribs. 

Observ. 

For  Movable  Ribs. 

5 

19 

.000000044 

23 

.000000122 

7 

27 

.000000208 

26 

.000000569 

9 

27 

.000000692 

50 

.000001900 

12 

24 

.000002801 

16 

.000011830 

The  computed   results   agree   very  well  with    the   observed 


21 

values,  except  for  small  velocities,  which,  however,  do  not  occur 
in  the  application  for  the  footstep.  The  value  of  ft  is  a  func- 
tion of  d.  To  establish  an  empirical  relation,  I  endeavored  to 
obtain  it  in  the  form  /?=A  ?/'  (<i),  in  which  '/'  (d)  is  a  simple 
known  function  of  d,  and  A  a  variable  coefficient  which 
changes  very  little  with  d,  so  that  its  proper  numerical  value 
may  be  obtained  with  sufficient  accuracy,  from  a  small  table. 
ft  was  found  to  be  well  represented  by  A  d5  ;  hence 


d 

Inches. 

VALUES  OF  /I  FOR 

Stationary  Ribs. 

Movable  Ribs. 

5 
7 
9 
12 

.000003503 
3080 
2915 
2801 

.000009713 
8415 
8007 
7690 

One-half  of  the  sum  of  these  values  of  A  for  stationary  and 
movable  wings  represents  the  true  value  of  A  for  our  step 
(see  page  10,  Resistance).  The  following  table  furnishes  inter- 
polated values  of  A  for  d  expressed  in  feet  : 


d 

Feet. 

A 

d 
Feet. 

A. 

.4 
.45 
.5 
.55 
.6 
.65 
.7 

.00000680 
.00000635 
.00000605 
.00000585 
.00000571 
.00000561 
.00000553 

.75 
.8 
.85 
.9 
.95 
1 

.00000546 
.00000541 
.00000535 
.00000531 
.00000528 
.00000525 

Recapitulation  of  1he  Results  for  Footstep. 

P=.001  d'n3m  neglecting  the  term  du 

d='  1 1000  p 

\    n  m 

L=.1047  n3hdsm=work  lost  by  friction  in  foot-pounds  per 
second. 


41H650 


22 

The  following  tables  offer  a  comparative  estimate  between 
the  hydraulic  step  and  those  in  common  use.  The  latter  have 
been  computed  with  a  view  to  durability,  from  the  data  given 
by  Reuleaux  in  the  fourth  edition  of  the  "  Konstrukteur  :  " 


HYDRAULIC    STEP. 


p 

n 

d 

m 

L 

2000 
2000 
2000 
2000 
8000 
8000 

2500 
2500 
400 
400 
2500 
1400 

0.75 
0.53 
1.88 
1.33 
075 
1.00 

1 
4 
1 
4 
4 
4 

2140 
1530 
780 
553 
8600 
6000 

FLAT  PIVOT.    STEEL  ON  BRONZE. 


P 

n 

Coefficient 
of 
Friction. 

( 
,     i 

Outer. 

1 

*  x 

Inner. 

L 

Lubri- 
cant. 

2000 

2500 

0.1 

1 

0.33 

17000 

8000 

2500 

0.1 

2 

0.66 

136000 

Oil. 

COLLAR  BEARINGS  ;  10  COLLARS.     STEEL  ON  BRONZE. 


2000 
8000 

2500 
2500 

.1 
.1 

0.25 
0.50 

015 

0.30 

5100 
40800 

Oil. 

PIVOT  OF  LIGNUM  VIT.E  ON  BRONZE. 


2000 

2500 

0.24 

0.13 

0 

5400 

2000 
8000 

400 
2500 

0.24 
024 

013 
0.26 

0 
0 

830 
43300 

Water 

8000 

1400 

0.24 

0.26 

0 

24200 

The  hydraulic  step  is  perfect  regarding  wear,  since  the  load 
is    completely    balanced    by    the    water   pressure   upon   the 


23 

disc.     This  superiority  is  also  maintained  with   reference  to 
to  the  loss  of  work  by  friction.     (See  tables. ) 


Fig.  11 


Fig.  11  represents  a  step   for  P=2000   and  ?i=2oOO,  and 
=9  inches. 

F.  G.  HESSE, 

PROFESSOR  OF  MECHANICAL  ENGINEERING. 

Berkeley,  July  15,  1887. 


UNIVERSITY  OF  CALIFORNIA 

AT 

LOS  ANGELES 
LIBRARY 


UNIVERSITY  OF  CALIFORNIA  LIBRARY 

Los  Angeles 
This  book  is  DUE  on  the  last  date  stamped  below. 


fit  C'D  LD-UKI 

MAY  2  9  1974 


Form  L9-Series  444 


UNIVERSITY  OF 
AT 

LOS  ANGELES 
LIBRARY 


